New Conditions for the Exponential Stability of Nonlinear Differential Equations

Hindawi Abstract and Applied Analysis Volume 2017, Article ID 4640835, 7 pages https://doi.org/10.1155/2017/4640835 Research Article New Conditions for the Exponential Stability of Nonlinear Differential Equations Rigoberto Medina Departamento de Ciencias Exactas, Universidad de Los Lagos, Casilla 933, Osorno, Chile Correspondence should be addressed to Rigoberto Medina; Received 23 January 2017; Accepted 23 March 2017; Published 10 April 2017 Academic Editor: Daoyi Xu Copyright © 2017 Rigoberto Medina. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We develop a method for proving local exponential stability of nonlinear nonautonomous differential equations as well as pseudolinear differential systems. The logarithmic norm technique combined with the “freezing” method is used to study stability of differential systems with slowly varying coefficients and nonlinear perturbations. Testable conditions for local exponential stability of pseudo-linear differential systems are given. Besides, we establish the robustness of the exponential stability in finite-dimensional spaces, in the sense that the exponential stability for a given linear equation persists under sufficiently small perturbations. We illustrate the application of this test to linear approximations of the differential systems under consideration. 1. Introduction The stability and robustness of differential systems have been widely investigated over the past decades; see, for example, [1–7] and references therein. This is due to theoretical interests and to being a powerful tool for system analysis and control design. The stability and robustness are the basic requirements for controlled systems. In practice, to satisfy the performance specification and to have a good transient response of the system, the controlled system is often designed to possess a stability degree. If the controlled system has a stability degree 𝛼, we say that the system is exponentially stable. The concept of 𝛼-stability is related to the exponential stability with a convergence rate 𝛼 > 0. Unlike the situation for linear systems, where necessary and sufficient conditions for stability are provided, the nonlinear problem is not completely solved. In fact, in spite of recent efforts (see [8–13] and the references therein), the exponential stability problem of nonlinear nonautonomous systems can be considered largely open. The main technique to stability of differential systems is Lyapunov’s method and its variants (Razumikhin-type theorems, LyapunovKrasovskii functional techniques); see, for example, [10, 14– 16]. In contrast, many alternative methods to Lyapunov’s functions have been successfully applied to the stability analysis of differential systems, for example, Ngoc [11], assuming that a nonlinear differential system with time-varying delay is bounded above by a positive linear time-invariant differential system and if this last system is exponentially stable then the nonlinear system under consideration is also exponentially stable. Anderson et al. [17], using the concept of Lyapunov exponents and Bohl exponents, discuss the problem of stabilization for linear time-varying systems with bounded matrices. Coppel [15], using the concept of ordinary and exponential dichotomy, establishes new results in stability theory, and also the “freezing” method became a fruitful tool among those alternative approaches; see, for example, Vinograd [18] and Gil and Medina [19]. In particular, the latter has been applied to prove that exponential stability of linear time-invariant differential systems implies the exponential stability of the system under consideration, provided that the coefficients of the original differential system are slowly varying. Moreover, an important tool to obtain explicit stability criteria for linear differential systems is the logarithmic norm of matrices (measure of matrices), which were used effectively in the recent literature on investigations of equations with dissipative coefficient matrices and their perturbations; see, for example, Zevin and Pinsky [12]. Besides, the logarithmic norm has been used to study the error bounds in the numerical integration of 2 Abstract and Applied Analysis ordinary differential equations [20, 21], estimates or stability of differential equations [15], and the oscillatory behavior of retarded functional differential equations [22]. Pseudo-linear systems are an important class of nonlinear systems. The stability and robustness of pseudo-linear differential equations are considered, for example, in [8, 10, 23–25]. Banks et al. [8] and Martynyuk [25] derived new bounds for solutions of perturbed pseudo-linear differential equations, basically using Gronwall-type inequalities. Dvirnyi and Slyn’ko [23, 24], constructing a piecewise differential Lyapunov function, established the stability of solutions to impulsive differential equations with impulsive action in the pseudo-linear form. Banks et al. [8], using a Gronwall-type inequality and assuming that a matrix 𝐵(𝑥, 𝑡) satisfies a jointly Lipchitz inequality in 𝑥 and 𝑡, established the robust exponential stability of evolution differential equations of pseudolinear form. In summary, in the existing literature there are many results concerning the stability or asymptotic behavior of pseudo-linear differential equations; however, in general, the assumptions are difficult to check or conservative. The purpose of this paper is to establish explicit conditions for the exponential stability of nonlinear differential systems. This approach led to study special classes of control systems, for example, systems with linear compact operators. In fact, assuming appropriate conditions on the perturbation term, the exponential feedback stabilization of a class of time-varying nonlinear systems can be established, provided the rate of variation of the system coefficients operators is sufficiently small. In this paper we consider differential systems defined in Euclidean spaces, with bounded operators on the right-hand side represented in the pseudo-linear form. New estimates for the norms of solutions are derived giving us explicit stability and boundedness conditions. The equations will be represented as a perturbation about a fixed value of the coefficient operator. Thus, applying norm estimates for the involved operator-valued functions, new stability results are established. The structure of this paper is as follows: in Section 2, we introduce some notations, the concept of stability with respect to a ball, and the definition and its properties of the logarithmic norm functions. In Section 3, the main exponential stability results and its consequences are established for nonlinear differential equations. In Section 4, we extend the main results to pseudo-linear differential (...truncated)